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Full text of "Analytical Mechanics"

ANGULAR IMPULSE AND ANGULAR MOMENTUM    267
210. Torque and Angular Momentum.—When the moment of inertia of a body remains constant under the action of a torque we have
(III)
Therefore torque equals the time rate of change of momentum.
The following analysis proves that the last statement is true when the moment of inertia varies with the time as well as when it remains constant.
Let Ay Fig. 122, represent a body, or a system of bodies, which is acted upon by one or more external torques. For the sake of simplicity suppose the planes of the torques to be parallel to the plane of the paper, and the axis of rotation to pass through the point 0 and to be perpendicular to the plane of the paper. Let dF be the resultant force acting upon an element of mass dm. Then the moment of dF about the axis of rotation equals the product of r, the distance of dm from the axis, by dFpJ the component of dF perpendicular to r. Therefore
dG = r - dFp = r - dmfp
= r.dm.l-~ (r2*)      [p. 97]
FIG. 122.
r dt
• dm •
— (r at
d dt